To solve the given differential equation:

$(x^2 + y^2 + 1)dx - (xy + y)dy = 0$

Step 1: Check if the equation is exact

The general form of an exact equation is: $M(x, y)dx + N(x, y)dy = 0$ where $M = x^2 + y^2 + 1$ and $N = -(xy + y)$.

To check for exactness, compute: $\frac{\partial M}{\partial y} = \frac{\partial}{\partial y}(x^2 + y^2 + 1) = 2y$ $\frac{\partial N}{\partial x} = \frac{\partial}{\partial x}(-(xy + y)) = -y$

Since $\frac{\partial M}{\partial y} \neq \frac{\partial N}{\partial x}$, the equation is not exact.

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Step 2: Find the integrating factor

We try to identify an integrating factor. An often-used integrating factor for such equations is a function of $x$ or $y$. Testing for $\mu(y) = y^{-1}$, multiply through by $\mu(y)$:

$\frac{1}{y}[(x^2 + y^2 + 1)dx - (xy + y)dy] = 0$

This simplifies to: $\left(\frac{x^2}{y} + y + \frac{1}{y}\right)dx - \left(x + 1\right)dy = 0$

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Step 3: Check if the new equation is exact

Now let: $M = \frac{x^2}{y} + y + \frac{1}{y}, \quad N = -(x + 1)$

Compute: $\frac{\partial M}{\partial y} = \frac{\partial}{\partial y}\left(\frac{x^2}{y} + y + \frac{1}{y}\right) = -\frac{x^2}{y^2} + 1 - \frac{1}{y^2}$ $\frac{\partial N}{\partial x} = \frac{\partial}{\partial x}(-(x + 1)) = -1$

Since $\frac{\partial M}{\partial y} \neq \frac{\partial N}{\partial x}$, the equation is still not exact.

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Step 4: Alternative integrating factor

A more general form of integrating factor might be a function of both $x$ and $y$. If $\mu = \frac{1}{x^2 + y^2 + 1}$, multiplying the original equation by this factor yields:

$\frac{1}{x^2 + y^2 + 1}[(x^2 + y^2 + 1)dx - (xy + y)dy] = 0$

Simplifying: $dx - \frac{xy + y}{x^2 + y^2 + 1}dy = 0$

Now, this equation can be integrated directly.

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Step 5: Solve the simplified equation

Rewrite as: $\frac{dy}{dx} = \frac{xy + y}{x^2 + y^2 + 1}$

This is separable if simplified further or solved using substitution methods depending on the specific context.

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Would you like me to proceed with a substitution, or clarify any steps so far?

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Related Questions:

1. What is the general process for checking exactness in differential equations?
2. How do we determine an appropriate integrating factor for a given equation?
3. Can this equation be solved directly without an integrating factor?
4. What substitution techniques are commonly used in equations involving $x^2 + y^2 + 1$?
5. How can symmetry in the equation help simplify the solving process?

Tip:

For non-exact equations, always test simple integrating factors ($x$-dependent, $y$-dependent, or proportional to a function of both $x$ and $y$) before resorting to more complex transformations.